On January 1, 2017, Smith deposits 1,000 into an account earning a nominal annual interest rate of i(4) = 0.04 compounded quarterly with interest credited on the last day of March, June, September, and December. If Smith closes the account during the year, the simple interest of 4% is paid on the balance from the most recent interest credit date.

(a) What is Smith’s close-out balance on September 23, 2017?

(b) Suppose all four quarters of the year are considered equal, and time is measured in years. Derive expressions for Smith’s accumulated amount function A(t), the close-out balance at time t. Consider separately the four intervals 0 ≤ t ≤ 0.25, 0.25 ≤ t ≤ 0.50, 0.50 ≤ t ≤ 0.75 and 0.75 ≤ t ≤ 1 and draw the timing diagram for each of these cases.

(c) Using part (b), show that if 0 ≤ t ≤ 0.25, then it follows that δt = δt+0.25 = δt+0.50 = δt+0.75.